PhD

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Space Spectral Time Fractional Finite Difference Method along with Stability Analysis for Fractional Order Nonlinear Wave Equations
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2018) Muhammad Sarmad Arshad; SP13-PMATH-004; Dr. Ayesha Sohail; LHR TP 5299
In this work, nonlinear partial differential equations governing the obscure phenomena of shallow water waves are discussed. Time fractional model is considered to understand the upcoming solutions on the basis of all historical states of the solution. A semi-analytic technique, Homotopy Perturbation Transform Method (HPTM) is used in conjunction with a numerical technique to validate the approximate solutions. With the aid of graphical interpretation, the favorable wave parameters, to avoid wave breaking are estimated. Afterwards, dynamical analysis of fractional order Schr dinger equation governing the optical wave propagation is reported in detail. The validity criteria for the application of the semi-analytic asymptotic methods are exploited. Comparison between the solutions obtained by the two asymptotic techniques, that is, the Fractional Homotopy Analysis Transform Method and the Optimal Homotopy Analysis Method is performed to select the most accurate technique for the stated problem. Space spectral analysis with integrating factor technique and time fraction finite difference method have been implemented to study the pressure waves propagating in bubbly fluids as well as nonlinear phenomena of plasma waves. Dynamical analysis of acoustic/pressure waves propagating in bubbly fluids is of great significance. Such flows arise in many engineering problems including sonochemistry, sonochemical reactors, cavitation around hydrofoils and ultrasonic propagation in medicine and biology. Fractional approach for modeling the propagation of the pressure waves in liquids containing a large number of tiny gas bubbles is proposed. Moreover, numerical solution of the fractional order Modified Korteweg-de Vries equation governing the dynamics is approximated using a novel space spectral time fractional finite difference tool. A spectral technique for space and a multi-step finite difference scheme for time are designed and implemented. The spatial spectral discretization error and the stability bounds are discussed. The nonlinear phenomena of plasma waves are well demonstrated with the aid of graphical analysis. Stability analysis of integer and fractional order KdV equations have been discussed quantitatively with the help of Evans function approximation.
New Measures of Intuitionistic Inclusion and Similarity with Applications
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2018) Madiha Qayyum; SP13-PMATH-002; Dr. Samina Mazhar; LHR TP 4873
Similarity and inclusion are two most important conceptual ways for looking at any possible relationships between objects (sets). A similarity measure is used for estimating the degree of resemblance/similarity between two objects while an inclusion measure expresses the degree to which one of the two is covered by the other one. In this dissertation, we have presented two different yet equally important approaches of defining similarity and inclusion measures for intuitionistic fuzzy sets. The first approach can be regarded as a logical approach implanting intuitionistic fuzzy implication, bi-implications and t-norms while the second one is a purely set theoretic approach having cardinalities and sets operations involved in its construction. In the logical approach, the degree of inclusion/ similarity was obtained by composing the intuitionistic logical operators (implications and bi-implications) with fuzzy measures on intuitionistic fuzzy sets. For this purpose we initially defined some new normal fuzzy measures on intuitionistic fuzzy sets along with a class of scalar cardinality measure for intuitionistic fuzzy sets. Later, we introduced different classes of intuitionistic fuzzy bi-implication operators having axiomatic as well as constructive approaches. A study on the properties of these bi-implication operators by utilizing Lukasiewicz intuitionistic fuzzy implicator revealed some remarkable results. The intuitionistic fuzzy bi-implication operators along with new defined fuzzy measures gave rise to multiple classes of intuitionistic fuzzy bi-implicator based similarity measures for intuitionistic fuzzy sets. Also the same normal fuzzy measures were employed to obtain the degree of inclusion between two intuitionistic fuzzy sets when composed with intuitionistic fuzzy implication operators. The new implication based classes of inclusion and similarity measures fulfilled almost all of the universally accepted criteria’s. In the set theoretic approach, we employed one of the members of new introduced scalar cardinality of intuitionistic fuzzy sets to construct a four parametric family of cardinality based similarity and inclusion measures. Both of these parametric families, xi for different combinations of parameters, generated intuitionistic fuzzy versions of some of the famous crisp measures of time. Lastly, the utility of the new measures (similarity, inclusion, cardinality and others) of intuitionistic fuzzy sets introduced in this work is exhibited by utilizing them as a part of solution techniques to the problems arising in real life situations.

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